A workbench for the reduction of detailed chemical kinetic

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A workbench for the reduction of detailed chemical kinetic mechanisms based on DRG and its deduced methods: Methodology and n-cetane as an example Yue Qiu, Liang Yu, Leilei Xu, Yebing Mao, and Xingcai Lu Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b00981 • Publication Date (Web): 31 May 2018 Downloaded from http://pubs.acs.org on May 31, 2018

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Energy & Fuels

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A workbench for the reduction of detailed chemical kinetic mechanisms based on DRG

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and its deduced methods: Methodology and n-cetane as an example

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Yue Qiu, Liang Yu, Leilei Xu, Yebing Mao, Xingcai Lu*

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Key Lab. for Power Machinery and Engineering of M. O. E, Shanghai Jiao Tong University, 200240, Shanghai, PR China

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Abstract:

Reduction of detailed mechanisms of large hydrocarbons is of significant importance to multi-CFD

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simulations, while the scale and accuracy of the reduced mechanism are closely related to the reduction method adopted. In

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this study, a workbench for systematic reduction of detailed mechanism was developed. It is operated on MATLAB

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platform and integrated with CHEMKIN PRO software. In the scheme, a skeletal reduction module was firstly employed to

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identify and eliminate unimportant species and associate reactions, in which four different algorithms based on DRG and its

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deduced methods, including single DRG, single DRGEP, two-stage DRG and DRG with DRGEP were applied and

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compared to find the optimal solution for further reduction. Then a subsequent reaction sensitivity analysis module was

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implemented to eliminate less important reactions. The potential and feasibility of the proposed scheme were presented with

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an example of reduction of a recently proposed detailed n-cetane mechanism. In skeletal reduction, DRG with DRGEP was

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found to be the optimal one and finally selected for skeletal reduction. Within 10% error tolerance, a comprehensive

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reduced mechanism consisting of 521 species and 1623 reactions was generated with ~75% reduction of species and 80%

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reduction of reactions. The reduced mechanism was well validated against the detailed mechanism in the ignition delay time,

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temperature profiles and important species concentrations in 0-D homogeneous batch reactor, species evolution in the

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jet-stirred reactor and 1-D premixed laminar flame speeds over a wide range of pressures (10-40 bar), temperatures

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(680-1600 K) and equivalence ratios (0.5-1.5).

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Keywords: Mechanism reduction; Skeletal mechanism; Deduced DRGs; N-Cetane; Diesel Surrogate

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Corresponding author: E-mail address: [email protected]. Tel.: +86-21-34206039; Fax: +86-21-34205949. 1

ACS Paragon Plus Environment

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1. Introduction

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Computational Fluid Dynamics (CFD) simulation is an effective technique in the design and development of modern

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engines. The flow, spray and combustion phenomenon of fuel inside cylinder are investigated by numerically solving fluid

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flow equations coupled with the chemical mechanism of fuel combustion in CFD simulation. However, on the one hand, the

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realistic fuel is a complex mixture of hundreds of thousands of alkanes, alkenes and aromatics. Even if we can constitute a

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surrogate model with limited representative components to characterize its major physiochemical properties, still the

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mechanism is too large for simulation. For instance, a complete mechanism describing the pyrolysis, partial oxidation and

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combustion of hydrocarbon and oxygenated fuels developed by E. Ranzi et al1 contains 451 species and 17848 reactions. A

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detailed diesel fuel surrogate model with binary mixture of n-dodecane and m-xylene also consists of 2885 species and

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11754 reactions2. While such a large scale may provide more details of the underlying chemical interactions, it is almost

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impossible to apply them in multi-CFD simulations. On the other hand, the species and reactions vary a lot in the detailed

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mechanism, which will induce severe chemical stiffness problems. The situation gets worse when a larger mechanism is

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involved3. Therefore, the reduction of detailed mechanisms is essential for multi-CFD. In addition, mechanism reduction

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can distinguish critical species and reactions which contributes to a clearer and more thorough understanding of the reaction

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process.

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Set against this background, much effort has been dedicated to the development of mechanism reduction in recent

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years3. In general, there are two major categories of reduction methods: skeletal reduction and time-scale analysis reduction.

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Time-scale analysis methods include typical quasi-steady state assumption (QSSA), partial-equilibrium approximation

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(PEA), computational singular perturbation (CSP) analysis, intrinsic low dimensional manifold (ILDM) and so on. More

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information can be found in Lu’s3 review. Skeletal reduction methods focus on identifying and eliminating unimportant

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species and reactions. It is efficient and typically the first step of mechanism reduction. Among them the Directed Relation

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Graph (DRG) first proposed in 2005 by Lu and Law4 gains much concern recently. It uses a directed graph to map the

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coupled species, wherein the vertex represents species and a direct weighted edge quantifies the dependences of one species

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on another. A skeletal mechanism can be generated by carefully setting the threshold to remove unimportant species. 2


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Following DRG, many developments and improvements have been made: Luo et al.5 revised the quantification of species

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interaction with ‘max’ operator in replace of original ‘sum’ operator for better reduction of mechanisms with large isomers;

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Lu proposed a two-stage DRG reduction strategy6 which restarts the reduction after the first round of DRG implementation

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and DRGASA7 which performs sensitivity analysis to further reduce unimportant species. While DRG is efficient, concise

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and easy for implementation, it only focuses on the direct relations between two species within one reaction without

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considering that the influences will be weakened if two species are far from each other. To compensate for this,

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Pepiot-Desjardins suggested a geometric error propagation strategy and developed DRGEP method8 which takes into

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consideration the length of chemical paths between two candidate species. A path dependent coefficient is adopted to

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quantify species interactions. Niemeyer9 compared the reliability and effectiveness of different graph search algorithms used

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in DRGEP and found that Dijkstra’s algorithm generated more compact skeletal mechanisms than depth first search (DFS),

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breadth first search (BFS) and revised breadth first search (RBFS) algorithm. Subsequently, Dijkstra algorithm was applied

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in DRGEP and later Niemeyer and Sung10 combined DRGEP with sensitivity analysis to obtain DRGEPSA. Besides

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DRGEP, there are also many other methods based on DRG such as the path flux analysis (PFA)11, 12, the flux project tree

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method13 and the Jacobi based DRGEP14. To sum up, DRG and its deduced skeletal reduction methods have been

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researched heatedly in recent years due to its simplicity, effectiveness, and minimal requirement of knowledge with fuel

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chemistry.

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There have been several successful applications of DRG and its deduced methods in the reduction of single

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components such as n-alkanes13, 15-17, iso-alkanes10, 18 and esters19-21, as well as multi-component surrogate models like

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gasoline22-25, diesel26-29, jet fuel30 and biodiesel5, 31, 32. In constructing surrogate models, hydrocarbons with large molecules

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are deemed as important components to characterize the physicochemical properties of target fuel. Among them, n-cetane is

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a favorable candidate in diesel surrogate models33, 34. First, it has comparable average carbon number with that of real diesel

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(C10-C22). In addition, it is a primary reference fuel for cetane number (CN) rating which permits fuel blending to adapt to

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the wide cetane-number range of real diesel. However, the chemical kinetic mechanism of n-cetane is not comprehensively

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studied owing to its complex long-chain structure. Westbrook et al.35 built a detailed mechanism with 2116 species and 8130 3


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reactions for combustion of n-alkanes (C8-C16)35 based on the classification of 25 reaction classes. Later, Sarathy et al.36

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made some modifications to the above n-alkane mechanism35 in their combined mechanism of C7-C20 n-alkanes and singly

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methylated alkanes. Those updates were demonstrated to increase low temperature reactivity, as validated by the authors.

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Researches on the reduction of detailed n-cetane mechanism have also been rare. Liang et al.37 generated a reduced

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381-species n-cetane baseline mechanism from Westbrook’s35 detailed mechanism using CHEMKIN-PRO Workbench to

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test advanced solution strategies in HCCI combustion, whilst the reduction details were not elaborately explained. Poon et

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al.34 proposed a five-stage reduction scheme and implemented it with the example of n-cetane. a reduced mechanism with

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79 species was derived from detail mechanism of Westbrook et al35 and validated in the 0-D closed homogeneous batch

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reactor and 2-D spray combustion simulation in a constant volume bomb. Nonetheless, the maximum error adopted is too

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large, reaching 40%. On the whole, large hydrocarbons such as n-cetane feature high boiling temperature, low saturate

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pressure and homogenously mixing with oxidizer. This leads to difficulties in experimental studies of ignition, oxidation,

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pyrolysis as well as laminar flame, and limits the development of detailed mechanism and subsequent reduction research.

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This paper attempts to further explore the feasibility of systematic reduction of detailed mechanism of large

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hydrocarbons. A reduction workbench was constructed based on MATLAB 2017b platform and integrated with

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CHEMKIN-PRO 1513138. It is composed of a skeletal reduction module followed by a reaction analysis module. In the

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skeletal reduction module, four different DRG-based algorithms, including single DRG, single DRGEP, two-stage DRG and

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DRG with DRGEP were discussed and compared to select an optimal method for further reduction. As an illustration, a

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reduced mechanism of n-cetane was generated from the newly constructed detailed mechanism by Sarathy et al.36 following

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the scheme and validated in the 0-D homogeneous batch reactor, jet stirred reactors (JSR) and 1-D premixed laminar flame

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speed simulator.

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2. Mechanism reduction workbench

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The reduction scheme proposed in this work focuses on skeletal reduction of detail mechanisms. An overall flow chart

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of the reduction process is illustrated in Fig.1. The original detailed mechanism, desired range of conditions (temperature,

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pressure and equivalence ratio) and the desired accuracy of the reduced mechanism are required inputs. 4


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First, CHEM is called to interpret the original detailed mechanism on MATLAB platform. Species and reactions are

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numbered and the thermal data and kinetic reactions are sorted in corresponding order. A matrix containing stoichiometric

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coefficients of species in the reactions is also generated. In this stage, the species and reactions are indexed by number so

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that any desired mechanism can be generated with given numbers.

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After that, constant volume simulations of autoignition within desired range of conditions are performed with

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CKReactorGenericClosed solver to sample ignition delay times and net reaction rates data. As current reduction focuses on

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the autoignition characteristic of fuels, the accuracy of the generated reduced mechanism refers to the maximum absolute

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relative error in ignition delay time prediction between the reduced mechanism and detailed mechanism. Herein, the ignition

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delay time is defined as the timepoint of the maximum temperature gradient along the temperature profile.

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Next, skeletal reduction module is implemented to identify and eliminate unimportant species. Sampled net reaction

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rates are used to calculate coupling coefficients between species. Thus, a weighted digraph is constructed where vertices

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denote species and edges denote the coupling of one species on another. Given a threshold, those edges with coefficients

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smaller than this value are removed from the graph while the remained edges connect the strongly coupled species. With a

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specific graph search method, a set of coupled important species can be identified and a skeletal mechanism containing only

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those selected species and associated reactions is generated. The desired skeletal mechanism which has the smallest scale

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under given accuracy is achieved by iteratively varying the thresholds and calculating the accuracy of generated skeletal

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mechanism. Four different skeletal reduction methods are implemented and compared, including single DRG, single

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DRGEP, two-stage DRG and DRG with DRGEP. DRG4 and DRGEP8 methods differ in calculation of species coupling

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coefficients and graph search algorithm. Herein, DRG method considers the net production of a species in calculating

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species coupling coefficients and adopts a depth first search algorithm to find a set of species coupled with initial target

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species. While in DRGEP reduction, the production and consumption of a species are considered separately and Dijkstra

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algorithm is applied to calculate the coupling coefficient between two species. One concern of implementation of DRG and

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DRGEP methods is that error can be induced after the elimination of some species due to the change of the mechanism

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coefficient matrix. Hence, multi-stage reduction is also considered while studies found that a two-stage reduction is 5


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adequate32. Therefore, single DRG, singe DRGEP, two-stage DRG and DRG with DRGEP are adopted and discussed at this

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stage to select the optimal one for further reduction.

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Once DRG-based skeletal reduction efficiently identifies and removes a large number of unimportant species, a

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reaction sensitivity analysis module is implemented to further eliminate less important reactions in the skeletal mechanism.

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Herein, a simple and reliable method is proposed in which reactions are ranked in ascending order of maximum reaction

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rates along evolution among all the sampling points. They are removed one by one until the accuracy of generated

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mechanism meets error limit. Note that a reversible reaction should be treated as a single reaction in case of fast reversible

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reactions that quickly reach partial equilibrium7.

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Following this scheme, a reduced mechanism with considerably fewer species and reactions is finally generated.

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Further external validations in the Jet Stirred Reactor(JSR), Laminar Flame Speed Calculator and so on are also conducted

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to test the accuracy and applicability of the generated reduced mechanism.

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3. Reduction of a detailed n-cetane mechanism

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3.1 Data sampling

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Recently, Sarathy et al.36 have made some updates to the n-alkane mechanisms of Westbrook et al.35 in construction of

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a detailed mechanism of C7-C20 2-methylalkanes (7200 species and 31400 reactions). Specifically, a new n-heptane

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mechanism by Mehl39,

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elementary reactions. Moreover, low-temperature reactivity of the mechanism was increased by a decrease in the activation

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energy of the alkyl peroxy radical isomerization class and the addition of a new concerted elimination reaction class. In this

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example, a sub-mechanism of n-cetane is generated from the above updated mechanism with 2136 species and 8076

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reactions (methyl-alkanes excluded) to serve as the initial detailed mechanism for reduction.

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was adopted as the core-mechanism with some adjustments of kinetic parameters of two

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Mechanism reduction is usually targeted at selected conditions. In this example, wide engine-relevant conditions were

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considered in order to ensure good applicability of the reduced mechanism. Pressures of 10, 20 and 40 bar, equivalence

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ratios of 0.5, 1.0 and 1.5, and a temperature range of 680-1600 K were selected. Within the temperature range, about 10 6


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temperature points were selected to cover low, NTC and high temperature regions, as is illustrated in Fig.2. Kinetic data

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were sampled at ~100 timepoints along temperature or pressure evolution profile (Fig.3). Denser sampling was made where

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temperature or pressure profile undergoes sharp rise to capture more information of the reaction.

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3.2 Comparison of different skeletal reduction methods

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Four skeletal reduction methods, including single DRG, single DRGEP, two-stage DRG and DRG with DRGEP were

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compared in the proposed workbench. They are different combinations of DRG and DRGEP algorithm, single stage and

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two-stage methods. In DRG method, fuel, O2, H2O and CO2 were selected as initial target species. In DRGEP method,

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careful selection should be made because the result is very sensitive to initial target species such as HO2, CO and CH2O

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near NTC regions for fuels with two-stage ignition22. In this example, fuel, O2, H, CO, HO2 and CH2O were selected. For

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the sake of simplicity, a preliminary skeletal reduction of n-cetane was conducted at a pressure of 20 bar, an equivalence

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ratio of 1, and a temperature range of 680-1600 K to illustrate the differences among the four methods. The desired accuracy

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of the generated skeletal mechanism is set at around 10%.

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Fig.4 compares the maximum error and species number at increasing thresholds of the investigated four methods.

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Fig.4(a) presents the result with single DRG method. As threshold increases, the species number of generated skeletal

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mechanism gradually decreases, while the maximum error of ignition delay time changes nonlinearly. It remains almost

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unchanged until the threshold increases to 0.1, after which the error steeply increases and then meets a trough as the

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threshold reaches 0.16. This indicates it is possible to generate higher accuracy of skeletal mechanism at an even smaller

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scale. Tosatto et al.41 found similar nonlinearity change of error curve in the reduction of JP-8 jet fuels. One reasonable

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explanation is that the calculated coefficient is an absolute value, not with sigh. It does not distinguish between the

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promotion and prohibition effects of one species on another. For example, the elimination of some species can reduce the

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error of predicted ignition delay time, while subsequent elimination of other species may increase the error. Thus, it is

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possible that the overall accuracy becomes higher. The nonlinear change of maximum error with increasing thresholds

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indicates that there probably exists a global optimal threshold for the reduction. In this case, considering the given 10%

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error tolerance, 0.11 was selected as the optimal threshold and 921 species were remained in skeletal mechanism with a 7


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maximum error of 12.89 %.

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Fig.4(b) demonstrates the case of single DRGEP method. As chemical path is considered, DRGEP can generate a much

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smaller scale of skeletal mechanism than DRG at a given threshold. The species number generally declines as the threshold

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increases, which undergoes a rapid decrease at the beginning and then slows down at a threshold of around 0.015. In

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contrast, the maximum error of predicted ignition delay undergoes fluctuation as threshold increases. The final optimal

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threshold was selected as 0.044 and a skeletal mechanism containing 511 species and 2061 reactions was generated. The

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maximum error was 10.86%.

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As for two-stage reduction methods, by setting a threshold of 1st DRG as 0.1(Fig.4(a)), a preliminary skeletal

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mechanism was generated and subsequent 2nd reduction was carried out with the method of DRG and DRGEP, respectively.

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Fig.4(c) and Fig.4(d) are plots of species number and maximum error of predicted ignition delay time against different

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thresholds in 2nd DRG and 2nd DRGEP methods. As can be seen from Fig.4, the curves of species number and maximum

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error against thresholds in 2nd reduction exhibit similar tendency to those of single stage reduction. Besides, the global

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optimum thresholds for maximum error are almost the same in the two stages, although the mechanisms are not as it was.

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This demonstrates the consistency of reduction algorithm on different reduction stages of mechanisms. In 2nd DRG

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reduction, the maximum error goes through a sharp rise at a threshold of ~0.14, at which point the maximum error is ~3%.

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However, further reduction of species will make the maximum error rapidly jump over 10% setting value. This possibly

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implies the strongly coupling of many species at that point. Finally, 0.138 was selected to generate a reduced mechanism

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with 777 species and 3.23% maximum error. As for 2nd DRGEP, 0.04 was selected as the threshold and the generated

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skeletal mechanism contains 518 species with 9.19% maximum error. Table 1 lists the detailed information of the skeletal

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mechanisms generated by four methods.

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Fig.5 depicts the maximum error of n-cetane skeletal mechanism versus species number so as to compare more

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intuitively the performances of four skeletal reduction methods. Overall, the DRG and its deduced methods can effectively

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reduce the scale of detailed mechanism at a small maximum error. In the example of n-cetane skeletal reduction, the

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maximum error is even less than 1.5% with ~50% reduction of species. As the species continue to be removed, the 8


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maximum error of generated skeletal mechanism raises rapidly and then goes through a trough. Similar tendency is found in

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all the four methods, indicating nonlinear change of maximum error during reduction process. In comparison with DRG,

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DRGEP algorithm can reach a larger reduction scale and generate a globally smaller mechanism within setting accuracy,

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although it shows big deviations at initial reduction stage. Besides, the implementation of a second stage reduction does

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have improved the reduction performance in this example. A larger number of species can be eliminated in the beginning

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and the maximum error curves are less fluctuating. Taken together, the two-stage reduction, DRG with DRGEP, was finally

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selected as the optimal method for skeletal reduction.

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3.3 Comprehensive reduction of n-cetane

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With the implementation of DRG with DRGEP method, skeletal reduction was carried out at all the sampled conditions

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covering a wide range of temperatures, pressures and equivalence ratios. The ultimate skeletal mechanism was generated by

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union of the retained species set at each case. It’s worth mentioning that mechanism reduction in this case exhibits little

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sensitivity to pressure and equivalence ratio. The reduced mechanism generated at P=20 bar and ER=1 also showed good

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performance at all other sampled conditions, with an overall maximum error being 11.11%. This is quite interesting as it

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indicates that the oxidation of n-cetane under different conditions can be characterized by simply some critical reactions,

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which in turn confirms the feasibility of mechanism reduction.

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After skeletal reduction, reaction sensitivity analysis was performed to further identify and eliminate unimportant

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reactions. Ranked reactions were removed iteratively, as explained in Section 2. Fig.6 displays the maximum error curve (all

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sampled conditions included) versus reaction number. With the removing of the reactions, the maximum error of the

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reduced mechanism undergoes a gradual decline in the beginning and then rises steeply as the reaction number approaches

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1620. This is interesting because the elimination of some unimportant species can actually increase the overall reduction

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accuracy. Finally, 411 reactions were removed from 2034 reactions under given 10% accuracy and the optimal mechanism

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contains 521 species and 1623 reactions, as is attached in Supplementary material. The overall maximum error decreases to

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8% after the unimportant reaction elimination.

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Fig.7 plots in detail the relative error of ignition delay times at each sampled condition. Similar tendency of relative 9


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error evolution with increasing temperature is found at different pressures and equivalence ratios. Overall, the reduced

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mechanism has agreeable accuracy at low and relatively high temperature (~1200 K), while large deviation occurs in NTC

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(~850-950 K) and higher temperature (>1400 K) region. Meanwhile, it can be concluded that the error is more sensitive to

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temperature than pressure and equivalence ratio.

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3.4 Validation of the reduced n-cetane mechanism

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Validation was performed via comparing the predicted ignition delay times of both the reduced mechanism and

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detailed mechanism in 0-D closed homogenous batch reactor within reduction range, as is shown in Fig.8. From the figure,

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the reduced mechanism performs well within a pressure of 10-40 bar, a temperature of 600-1600 K and an equivalence ratio

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of 0.5-1.5. The results of the reduced mechanism almost overlap with those of the detailed mechanism. Even the deviation

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in NTC region where the largest error occurs is narrow from the figure. Within 10% error tolerance, the generated reduced

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mechanism features ~75% reduction of species and ~80% reduction of reactions.

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Existing studies23, 24 found that mechanism reduction targeted at ignition delay time prediction cannot guarantee the

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prediction of time and spatial evolution of some parameters. Hence, temperature and major species evolution were also

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validated in this work, as is shown in Fig.9-10. Three initial temperature points (T=730 K, 910 K and 1200 K) at P=20 bar

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and ER=1 were selected to represent low, NTC and high temperature regions, respectively. It is revealed from Fig.9 that the

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reduced mechanism overall predicts well the temperature and OH evolution, with a bit shift towards left at NTC point and

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right at low temperature point. Similar tendency is also found in Niemeyer’s10 reduction of n-heptane and n-decane.

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Meanwhile, two-stage ignition of n-cetane at low and medium temperature is well captured by the reduced mechanism.

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Fig.10 further compares other mole fraction evolution of major species, wherein NC16H34, O2 and CO2 are major

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reactants and products, HO2 is an important species in chain branching reactions and CH2O is critical in the formation of

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CO. In correspondence with temperature and OH profile, the reduced mechanism performs well at high temperature, while

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there exist some deviations at NTC point. The mole fraction evolution curves are found to be shifted left in the reduced

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mechanism, which means NC16H34 was consumed earlier to initiate low-temperature reactions. Temperature rose earlier

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and the overall reaction process was advanced. This is possibly due to the elimination of some chain-inhibiting reactions at 10


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Energy & Fuels

low temperature.

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Extended validation of the reduced mechanism was also performed in jet stirred reactor(JSR) at P=10 atm, ER=1 with

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residence time being 0.1s. Mole fractions of major species are shown in Fig.11. It is demonstrated that the reduced

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mechanism reproduces well the species concentration at different temperatures, even in low temperature region. The

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nonmonotonic change of the species mole fraction as temperature rises is also observed, which can be explained by the

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NTC behavior of the fuel. Meanwhile, 1-D simulation of premixed laminar flame speeds was also conducted using both the

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detailed and reduced mechanism at 10-40 atm under an unburnt gas temperature of 400 K. The effects of thermal diffusion

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(the Soret effect) was taken into consideration and the mixture-averaged diffusion model was applied. There were around

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100 grid points in the solving domain and the final results of calculated laminar flame speeds were plotted in Fig.12. As can

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be seen from the figure, the reduced mechanism predicts perfectly well the laminar flame speeds as the equivalence ratio

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varies over 0.5-1.5, with nearly no deviations. The high temperature chemistry of the detailed mechanism is well captured

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by the reduced mechanism. This further confirms the feasibility of characterizing the oxidation of n-cetane with a

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small-scale reduced mechanism.

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4. Concluding Remarks

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A workbench for the systematic reduction of the detailed hydrocarbon mechanisms was proposed in this study. It was

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operated on MATLAB platform and integrated with CHEMKIN PRO software. In the scheme, a skeletal reduction module

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was firstly employed to identify and eliminate unimportant species and associate reactions, in which four different

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algorithms based on DRG and its deduced methods, single DRG, single DRGEP, two-stage DRG and DRG with DRGEP

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were applied and compared to find the optimal one for further reduction. After that, a reaction sensitivity analysis module

257

was implemented to eliminate less important reactions.

258

The potential and flexibility of the proposed scheme was presented with an example of the reduction of n-cetane,

259

which is a favorable diesel surrogate component. In the skeletal reduction module, all four methods can effectively reduce

260

the scale. The maximum error of predicted ignition delay time changes nonlinearly as threshold increases and there exists a

261

global optimum point. In comparison with DRG, DRGEP algorithm can reach a larger reduction scale and generate a 11


Energy & Fuels 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 27

262

globally smaller mechanism within setting accuracy, although it shows bigger deviations at initial reduction stage.

263

Two-stage methods can generate a reduced mechanism with a higher accuracy at the same species number than single-stage

264

methods. Overall, DRG with DRGEP was found to be the optimal one and finally selected as the skeletal reduction method.

265

Within 10% error tolerance, a comprehensive reduced mechanism consisting of 521 species and 1623 reactions was

266

generated with around 75% reduction of species and 80% reduction of reactions. The reduced mechanism was validated

267

against the ignition delay times, temperature profiles and important species concentrations in the 0-D homogeneous batch

268

reactor, species evolution in the jet-stirred reactor and laminar flame speeds in 1-D flame simulator over a wide range of

269

pressures (10-40 bar), temperatures (680-1600 K) and equivalence ratios (0.5-1.5). The results overall can well reproduce

270

the oxidation of n-cetane with narrow deviations. The performances in high temperature region are slightly better than those

271

of low temperature region and the maximum error usually lies in NTC region. Besides, it is found in the reduction of

272

n-cetane that mechanism reduction is less sensitive to pressure and equivalence ratio as the generated mechanism at one

273

condition also showed agreeable performances at full reduction ranges. In summary, the systematic reduction workbench

274

exhibits great feasibility and potential for large-scale reduction of detailed mechanisms.

275

Acknowledgements

276

This work was supported by the Natural Science Foundation of China (Grant No. 51425602, 91641202) and High

277

Technology Ship Research Program-Marine Low Speed Project (Phase I).

278

References

279 280 281 282 283 284 285 286 287 288 289 290 291

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2. 3. 4. 5. 6.

Ranzi, E.; Frassoldati, A.; Grana, R.; Cuoci, A.; Faravelli, T.; Kelley, A. P.; Law, C. K., Hierarchical and comparative kinetic modeling of laminar flame speeds of hydrocarbon and oxygenated fuels. Progress in Energy and Combustion Science 2012, 38, (4), 468-501. Pei, Y.; Mehl, M.; Liu, W.; Lu, T.; Pitz, W. J.; Som, S., A multicomponent blend as a diesel fuel surrogate for compression ignition engine applications. Journal of Engineering for Gas Turbines and Power 2015, 137, (11), 111502. Lu, T. F.; Law, C. K., Toward accommodating realistic fuel chemistry in large-scale computations. Progress in Energy and Combustion Science 2009, 35, (2), 192-215. Lu, T. F.; Law, C. K., A directed relation graph method for mechanism reduction. Proceedings of the Combustion Institute 2005, 30, 1333-1341. Luo, Z. Y.; Lu, T. F.; Maciaszek, M. J.; Som, S.; Longman, D. E., A Reduced Mechanism for High-Temperature Oxidation of Biodiesel Surrogates. Energy & Fuels 2010, 24, (12), 6283-6293. Lu, T. F.; Law, C. K., Linear time reduction of large kinetic mechanisms with directed relation graph: n-Heptane and iso-octane. Combustion and Flame 2006, 144, (1-2), 24-36. 12


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Energy & Fuels Lu, T. F.; Law, C. K., Strategies for mechanism reduction for large hydrocarbons: n-heptane. Combustion and Flame 2008, 154, (1-2), 153-163. Pepiot-Desjardins, P.; Pitsch, H., An efficient error-propagation-based reduction method for large chemical kinetic mechanisms. Combustion and Flame 2008, 154, (1-2), 67-81. Niemeyer, K. E.; Sung, C.-J., On the importance of graph search algorithms for DRGEP-based mechanism reduction methods. Combustion and Flame 2011, 158, (8), 1439-1443. Niemeyer, K. E.; Sung, C. J.; Raju, M. P., Skeletal mechanism generation for surrogate fuels using directed relation graph with error propagation and sensitivity analysis. Combustion and Flame 2010, 157, (9), 1760-1770. Sun, W. T.; Chen, Z.; Gou, X. L.; Ju, Y. G., A path flux analysis method for the reduction of detailed chemical kinetic mechanisms. Combustion and Flame 2010, 157, (7), 1298-1307. Wang, W.; Gou, X., An improved path flux analysis with multi generations method for mechanism reduction. Combustion Theory and Modelling 2016, 20, (2), 203-220. Liu, A.-K.; Jiao, Y.; Li, S.; Wang, F.; Li, X.-Y., Flux Projection Tree Method for Mechanism Reduction. Energy & Fuels 2014, 28, (8), 5426-5433. Chen, Y.; Chen, J.-Y., Application of Jacobian defined direct interaction coefficient in DRGEP-based chemical mechanism reduction methods using different graph search algorithms. Combustion and Flame 2016, 174, 77-84. Sung, C. J.; Niemeyer, K. E., Skeletal Mechanism Generation of Surrogate Jet Fuels for Aeropropulsion Modeling. Iscm Ii and Epmesc Xii, Pts 1 and 2 2010, 1233, 1412-1417. Bahlouli, K.; Saray, R. K.; Atikol, U., Development of a Reduced Mechanism for n-Heptane Fuel in HCCI Combustion Engines by Applying Combined Reduction Methods. Energy & Fuels 2012, 26, (6), 3244-3256. Yao, T.; Pei, Y. J.; Zhong, B. J.; Som, S.; Lu, T. F.; Luo, K. H., A compact skeletal mechanism for n-dodecane with optimized semi-global low-temperature chemistry for diesel engine simulations. Fuel 2017, 191, 339-349. Li, R.; He, G.; Zhang, D.; Qin, F., Skeletal Kinetic Mechanism Generation and Uncertainty Analysis for Combustion of Iso-octane at High Temperatures. Energy & Fuels 2018, 32, (3), 3842-3850. Gerasimov, I. E.; Bolshova, T. A.; Zaev, I. A.; Lebedev, A. V.; Potapkin, B. V.; Shmakov, A. G.; Korobeinichev, O. P., Reduced Chemical Kinetic Mechanism for Methyl Pentanoate Combustion. Energy & Fuels 2017, 31, (12), 14129-14137. Lin, K. C.; Tao, H.; Kao, F.-H.; Chiu, C.-T., Minimized Skeletal Mechanism for Methyl Butanoate Oxidation and Its Application to the Prediction of C3–C4Products in Nonpremixed Flames: A Base Model of Biodiesel Fuels. Energy & Fuels 2016. Liu, C.; Zuo, Z.; Feng, H., Skeletal and Reduced Chemical Kinetic Mechanisms for Methyl Butanoate Autoignition. Energy & Fuels 2016, 31, (1), 891-895. Chen, Y.; Wolk, B.; Mehl, M.; Cheng, W. K.; Chen, J.-Y.; Dibble, R. W., Development of a reduced chemical mechanism targeted for a 5-component gasoline surrogate: A case study on the heat release nature in a GCI engine. Combustion and Flame 2017, 178, 268-276. Niemeyer, K. E.; Sung, C. J., Mechanism reduction for multicomponent surrogates: A case study using toluene reference fuels. Combustion and Flame 2014, 161, (11), 2752-2764. Niemeyer, K. E.; Sung, C. J., Reduced Chemistry for a Gasoline Surrogate Valid at Engine-Relevant Conditions. Energy & Fuels 2015, 29, (2), 1172-1185. Wang, H.; Yao, M.; Reitz, R. D., Development of a Reduced Primary Reference Fuel Mechanism for Internal Combustion Engine Combustion Simulations. Energy & Fuels 2013, 27, (12), 7843-7853. Poon, H. M.; Pang, K. M.; Ng, H. K.; Gan, S.; Schramm, J., Development of multi-component diesel surrogate fuel models – Part I: Validation of reduced mechanisms of diesel fuel constituents in 0-D kinetic simulations. Fuel 2016, 180, 433-441. Poon, H. M.; Pang, K. M.; Ng, H. K.; Gan, S.; Schramm, J., Development of multi-component diesel surrogate fuel models – Part II: Validation of the integrated mechanisms in 0-D kinetic and 2-D CFD spray combustion simulations. Fuel 2016, 181, 120-130. 13


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28. Qiu, L.; Cheng, X.; Wang, X.; Li, Z.; Li, Y.; Wang, Z.; Wu, H., Development of a Reduced n-Decane/α-Methylnaphthalene/Polycyclic Aromatic Hydrocarbon Mechanism and Its Application for Combustion and Soot Prediction. Energy & Fuels 2016, 30, (12), 10875-10885. 29. Sun, X.; Liang, X.; Shu, G.; Wang, Y.; Wang, Y.; Yu, H., Development of a Reduced n-Tetradecane–Polycyclic Aromatic Hydrocarbon Mechanism for Application to Two-Stroke Marine Diesel Engines. Energy & Fuels 2016, 31, (1), 941-952. 30. Wang, Q.-D.; Fang, Y.-M.; Wang, F.; Li, X.-Y., Systematic analysis and reduction of combustion mechanisms for ignition of multi-component kerosene surrogate. Proceedings of the Combustion Institute 2013, 34, (1), 187-195. 31. Luo, Z. Y.; Plomer, M.; Lu, T. F.; Som, S.; Longman, D. E., A reduced mechanism for biodiesel surrogates with low temperature chemistry for compression ignition engine applications. Combustion Theory and Modelling 2012, 16, (2), 369-385. 32. Poon, H. M.; Ng, H. K.; Gan, S.; Pang, K. M.; Schramm, J., Evaluation and Development of Chemical Kinetic Mechanism Reduction Scheme for Biodiesel and Diesel Fuel Surrogates. SAE International Journal of Fuels and Lubricants 2013, 6, (3), 729-744. 33. Pitz, W. J.; Mueller, C. J., Recent progress in the development of diesel surrogate fuels. Progress in Energy and Combustion Science 2011, 37, (3), 330-350. 34. Poon, H. M.; Ng, H. K.; Gan, S.; Pang, K. M.; Schramm, J., Development and Validation of Chemical Kinetic Mechanism Reduction Scheme for Large-Scale Mechanisms. SAE International Journal of Fuels and Lubricants 2014, 7, (3), 653-662. 35. Westbrook, C. K.; Pitz, W. J.; Herbinet, O.; Curran, H. J.; Silke, E. J., A comprehensive detailed chemical kinetic reaction mechanism for combustion of n-alkane hydrocarbons from n-octane to n-hexadecane. Combustion and Flame 2009, 156, (1), 181-199. 36. Sarathy, S. M.; Westbrook, C. K.; Mehl, M.; Pitz, W. J.; Togbe, C.; Dagaut, P.; Wang, H.; Oehlschlaeger, M. A.; Niemann, U.; Seshadri, K.; Veloo, P. S.; Ji, C.; Egolfopoulos, F. N.; Lu, T., Comprehensive chemical kinetic modeling of the oxidation of 2-methylalkanes from C-7 to C-20. Combustion and Flame 2011, 158, (12), 2338-2357. 37. Liang, L.; Puduppakkam, K.; Meeks, E., Towards Using Realistic Chemical Kinetics in Multidimensional CFD. 2009. 38. Reaction Design: San Diego, CHEMKIN-PRO. 15131; 2013. 39. Mehl, M.; Pitz, W. J.; Sjöberg, M.; Dec, J. E., Detailed Kinetic Modeling of Low-Temperature Heat Release for PRF Fuels in an HCCI Engine. In SAE International: 2009. 40. Mehl, M.; Pitz, W. J.; Westbrook, C. K.; Curran, H. J., Kinetic modeling of gasoline surrogate components and mixtures under engine conditions. Proceedings of the Combustion Institute 2011, 33, 193-200. 41. Tosatto, L.; Bennett, B. A. V.; Smooke, M. D., Comparison of different DRG-based methods for the skeletal reduction of JP-8 surrogate mechanisms. Combustion and Flame 2013, 160, (9), 1572-1582.

14


Page 15 of 27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

Table 1 Comparison of skeletal mechanisms of n-cetane generated by different methods

Method

Thresholds

#Species

#Reactions

Single DRG Single DRGEP Two-stage DRG DRG+DRGEP

0.11 0.044 0.1, 0.138 0.1, 0.04

901 511 777 521

3736 2061 3079 2034

15


Max. Error /% 12.89 10.86 3.23 9.19

Energy & Fuels 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Fig.1. Flow chart of the reduction scheme

16


Page 16 of 27

Page 17 of 27

10

ignition delay (ms)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

TL=680K

1 0.1 0.01 1E-3

TH=1600K

0.6

0.8

P=20 bar, ER=1 T=1000K

1.0

1.2

1.4

1.6

1000/T (1/K) Fig.2. An illustration of data sampling of temperature points

17


Page 18 of 27

5000

60

4000

40

3000

20

0

2000

P=20 bar, ER=1 T=1000K

1000 0.0

0.5

1.0

1.5

2.0

2.5

3.0

-20 3.5

time (ms)

Fig.3. An illustration of data sampling of timepoints

18


Pressure (bar)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Temperature (K)

Energy & Fuels

(b)

(a) 0.6

0.3

1000 0.2

optimal point

0.1

0.5 1500

0.4 0.3

1000 0.2 0.1

500

nd

2 reduction

0.00

0.05

0.10

0.15

optimal point

0.0

0.00

0.20

0.03

0.04

0.05

0.0 0.06

1000 (c)

(d) 0.6

0.6

Number of Species

1000

0.4 0.3 600 0.2 0.1

optimal point 0.05

0.10

Maximum Error

0.5 800

0.00

0.02

1-DRGEP Threshold

1-DRG Threshold

400

0.01

0.5 800 0.4 0.3 600 0.2 0.1 400

0.15

0.20

Maximum Error

500

Number of Species

0.4

Maximum Error

Number of Species

0.5

1500

0.6

2000

2000

Number of Species

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

Maximum Error

Page 19 of 27

optimal point

0.0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.0 0.25

nd

nd 2 -DRGEP Threshold 2 -DRG Threshold Fig.4 Species number and maximum error of ignition delay time prediction against different thresholds for n-cetane skeletal mechanism generated by different methods (a) DRG, (b) DRGEP, (c)2nd DRG with 0.1 threshold at 1st DRG, (d)2nd DRGEP with 0.1 threshold at 1st DRG

19


Energy & Fuels

0.8 single DRG 2-stage DRG DRG+DRGEP single DRGEP

0.7 0.6

Maximum Error

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 27

0.5 0.4 0.3 0.2 0.1 optimal point

0.0 250

500

750

1000 1250 1500 1750 2000

Number of Species Fig.5 Comparison of the maximum error against species number of n-cetane skeletal mechanism generated by four methods.

20


Page 21 of 27

0.4

Maximum Error

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

0.3

411 reactions are removed from the skeletal mechanism; The final reduced mechansim contains 521 species and 1623 reactions

0.2

0.1

optimal point 0.0 1500

1600

1700

1800

1900

2000

Number of Reactions

Fig.6 Maximum error curve against reaction number

21


P=10, ER=0.5 P=10, ER=1.0 P=10, ER=1.5

600 800 1000 1200 1400 1600

T (K)

0.08 0.06 0.04 0.02 0.00 -0.02 -0.04 -0.06 -0.08 600

P=20, ER=0.5 P=20, ER=1.0 P=20, ER=1.5

800

1000 1200 1400 1600

T (K)

Page 22 of 27

Relative Error

0.08 0.06 0.04 0.02 0.00 -0.02 -0.04 -0.06 -0.08

Relative Error

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Relative Error

Energy & Fuels

0.08 0.06 0.04 0.02 0.00 -0.02 -0.04 -0.06 -0.08 600

P=40, ER=0.5 P=40, ER=1.0 P=40, ER=1.5

800

1000 1200 1400 1600

T (K)

Fig.7 Comparison of maximum error of reduced mechanism at all sampled conditions: (a)10 bar, (b)20 bar, (c)40 bar

22


Page 23 of 27

100

100 (a)

10

10 bar

1

20 bar 40 bar

0.1 0.01

10 10 bar

1

0.8

1.0

1.2

20 bar 40 bar

0.1 0.01

line:detailed symbol:reduced

1E-3 0.6

1.4

line:detailed symbol:reduced 0.8

1.0

1000/T (1/K)

1.2

1.4

1000/T (1/K)

100 (c)

ER:1.5

ignition delay (ms)

1E-3 0.6

(a)

ER:1.0

ignition delay (ms)

ER:0.5

ignition delay (ms)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

10 10 bar

1 20 bar 40 bar

0.1 0.01 1E-3 0.6

line:detailed symbol:reduced 0.8

1.0

1.2

1.4

1000/T (1/K) Fig.8 Comparison of ignition delay times of detailed and reduced mechanism within reduction range. (line: detailed, symbol: reduced) (a) ER=0.5, (b) ER=1.0, (c) ER=1.5

23


Energy & Fuels

3500

0.020

(a)

2500

Mole Fraction

3000

T (K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1200K 910K

2000 730K 1500

P=20bar, ER=1 line:detailed symbol:reduced

1000 500 0.0

0.5

1.0

1.5

2.0

2.5

Page 24 of 27

(b)

0.015

OH 1200K

910K

0.010

730K 0.005

0.000 0.0

P=20bar, ER=1 line:detailed symbol:reduced 0.5

1.0

1.5

2.0

2.5

time (ms) time (ms) Fig.9 Temperature and OH profiles in ignition delay time prediction with both the detailed and reduced mechanism at P=20 bar, ER=1 (line: detailed, symbol: reduced) (a)Temperature, (b) OH mole fraction

24


Page 25 of 27

0.010

1200K 730k

0.004 910K

P=20bar, ER=1 line: detailed symbol:reduced

0.002

Mole Fraction

Mole Fraction

0.006

O2

0.20

0.000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 time (ms)

0.15

1200K

0.10 910K 730k

0.05 0.00 0.0

0.5

1.0 1.5 2.0 time (ms)

0.10 Mole Fraction

NC16H34 0.008

1200K

0.05 910K 730k

0.00 0.0

2.5

CO2

0.5

1.0 1.5 2.0 time (ms)

2.5

0.0010 0.020

HO2

C2H4

1200K

0.010 910K

0.005 0.000 0.0

730k

0.5

1.0 1.5 2.0 time (ms)

2.5

0.0006

1200K

0.0004 910K 730k

0.0002 0.0000 0.0

0.5

1.0 1.5 time (ms)

2.0

Mole Fraction

Mole Fraction

0.015

CH2O

0.006

0.0008 Mole Fraction

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

0.004 1200K 910K

0.002 0.000 0.0

730k

0.5

1.0 1.5 2.0 time (ms)

2.5

Fig.10 Mole fractions of major species in ignition delay time simulation with both the detailed and reduced mechanism at P=20 bar, ER=1 (line: detailed, symbol: reduced)

25


Energy & Fuels

-3

-5

10

-7

10

-8

10

τ =0.1s P=20bar, ER=1 line: detailed symbol: reduced

Mole Fraction

-4

-6

-1

10

-1

10

10

10

O2

NC16H34 Mole Fraction

Mole Fraction

10

-2

10

-3

10

800 1000 1200 1400 1600 T (K)

800

1000 1200 1400 1600 T (K)

Mole Fraction

-3

10

-4

10

10

10

C2H4

-5

-3

10

800

1000 1200 1400 1600 T (K)

-2

10 Mole Fraction

-2

10

-2

10

10

1000 1200 1400 1600 T (K)

-2

CO

CO2

-3

800

-1

10 Mole Fraction

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 27

CH2O

-3

10

-4

10

-5

10

-6

800

1000 1200 1400 1600 T (K)

10

800

1000 1200 1400 1600 T (K)

Fig.11 Mole fractions of major species in JSR model simulation with both the detailed and reduced mechanism at P=10 atm, ER=1 and τ =0.1s (line: detailed, symbol: reduced)

26


Page 27 of 27

40

Laminar Flame Speed (cm/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

35

Tu=400K

30

10 atm

25

20 atm

20 15

40 atm

10 line: detailed symbol: reduced

5 0 0.4

0.6

0.8

1.0

1.2

1.4

1.6

ER Fig. 12 Premixed laminar flame speeds versus equivalence ratios predicted by both the detailed and reduced mechanism At Tu=400 K (line: detailed, symbol: reduced)

27


A workbench for the reduction of detailed chemical kinetic

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